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The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

1.  Introduction

Model predictive control or a receding horizon control
has become an attractive feedback strategy during recent
years. Generalized predictive control (GPC) of linear sys-
tems has found successful applications, especially in the
process industries, due to its robustness to parameter
uncertainties (Boucher and Dumar, 1996).   However,
many systems are, in general, inherently nonlinear. Thus,
linear models are often inadequate to describe the process
dynamics and nonlinear models have to be used. Much
effort has been made to extend GPC to nonlinear systems
(Michalska and Mayne, 1993).  However, in model pre-
dictive control, a nonlinear optimization problem must be
solved online with computational complexity (Henson
and Seborg, 1997). Therefore, this kind of control law can
be applied only for systems with slow dynamics (for
example chemical process). To avoid this, several nonlin-
ear predictive laws have been developed  (Ping, 1995;
Singh, et al. 1995; Souroukh and Kravaris, 1996) where
the one-step ahead predictive error is obtained by  expand-
ing the output  signal and reference signal in a rith order
Taylor series, and rith is the relative degree of the ith out-
put.  Then, the continuous minimization of the predicted
_________________________________________
*Corresponding author’s e-mail: hedjar@ccis.ksu.edu.sa

tracking errors is used to derive the control law. This paper
examines the nonlinear continuous-time generalized pre-
dictive control approach based on a finite horizon dynam-
ic minimization of predicted tracking errors with end
point constraints to achieve tracking objectives. It is
noticed that, the proposed approach increases the tracking
performance with regards to Ping's Lu method (Ping,
1995) and Chen's method (Chen, et  al. 2003).  Indeed, the
two performance indexes used by Ping (1995) and  Chen,
et al. (2003) are combined (Control Integration) to
increase the dynamic of the tracking error. Moreover, the
recent Chen et al. method given in Chen, et al. ( 2003)
assumes that decoupling matrix is not singular, which is
not practical. With the proposed combination control, the
assumption above will be weakened. It will be shown that
some advantages of this control scheme include good
tracking performance, clear physical meaning of maxi-
mum and minimum control values when saturation occurs
with regard to input-output linearization method.

Minimizing a predictive cost function derives the pro-
posed nonlinear predictive controller. A mobile robot is
used as an illustrative example to show the tracking per-
formance achieved by this nonlinear predictive controller.
Note that Taylor approximation is used to overcome the
computation burden. 

The rest of the paper is organized as follows: In section
2, the problem statement is given; a control law is devel-

Nonlinear Predictive Control With End Point Constraints
R. Hedjar*

Computer Engineering Department, College of Computer and Information Sciences - King Saud University,
P.O.Box 51178- Riyadh 11543, Saudi Arabia

Received 8 May 2005 ; accepted 16 November 2005

Abstract: The optimal nonlinear predictive control structure with end point constraints is presented, which provides asymp-
totic tracking of smooth reference trajectories. The controller is based on a finite horizon continuous time minimization of
nonlinear predicted tracking errors. A key feature of the control law is that its implementation does not need to perform an
online optimization, and asymptotic tracking of smooth reference signal is guaranteed.  The proposed control scheme is
applied to planning motions problem of a mobile robot. Simulations results are performed to validate the tracking perform-
ance of the proposed controller.

Keywords: Nonlinear predictive control, Stability, Mobile robot and tracking trajectory

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70

The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

oped to minimize the difference between predicted and
desired response. The properties of the control law are dis-
cussed, including stability. In section 3, the proposed con-
troller is applied to the planning motion problem of the
mobile robot.  The results are summarized in section 4
where some directions for related research are also pro-
vided.

2. Model Formulation

2.1 Optimal Nonlinear Predictive Control
In the receding horizon control strategy, the following

control problem is solved at each t > 0 and x(t) :

(1)

subject to the state equation:

(2)

On one hand, the objective function used by Ping  is
the one step predicted tracking error given by: 

(3)

On the other hand, the cost function used by Chen et al.
(2003) is the predicted tracking error over a fixed horizon
without control penalty, and is given by:

(4)

It is noticed that the derived optimal control of the
above cost function increases the dynamics of the tracking
error with regard to Ping's method, but the decoupling
matrix should be nonsingular. 

In this paper, to maximize the tracking performance of
previous methods and overcome the assumption of the
decoupling matrix, we propose the integration of these
two control structures (Control Integration), and the cost
function to be minimized is rewritten as:

(5)

where Q1 and Q2 are positive definite matrices respective-
ly and R positive semi-definite matrix.

(6)

A simple and effective way to predict the influence of
u(t) on x(t+h) is to use rith order Taylor series expansion,
in such a way as to obtain for each component of the vec-
tor:

(7)

The predicted state variable in compact form is given
by:

(8)

where 

We also expand each component of  xref (t+h) in rith order
Taylor series to have:

xref (t+h) = xref (t) + d(t,h)                                           (9)

where  d(t,h) = [d1 d2 ... dn]T with

[ ]τττ+ττ
=

∫
+

d)(uR)(u)(xQ)(x
2
1

Min)t),t(u),t(x(JMin
ht

t

TT

)t(u

)t(u)x(g)x(f)t(x +=&

and the terminal end point constraint x(t+h)  = 0 for 
some h > 0, where x(t) ⊂ X∈ ℜn is the state vector,  u(t) 
⊂ U∈ ℜm represents the control vector. The functions 
f(x): ℜn  → ℜn and g(x) : ℜ n  → ℜ n×m are sufficiently 
differentiable, Q is positive definite matrix  and R  
positive semi -definite matrix. We denote the optimal 
control vector to the above problem by: u*( τ), τ∈ [t, 
t+h]. The currently applied control u(t) is set equal to  
u*(τ). This process is repeated for every next t for 
stabilization of the system at  the origin. However, 
solving a nonlinear dynamic optimization problem with 
equality constraints is highly computationally intensive, 
and in many cases it is impossible to be performed 
within a reasonable time limit. Furthermore, the global 
optimization so lution cannot be guaranteed in each 
optimization procedure since, in general, it is a non -
convex, constrained nonlinear optimization problem 
(Henson and Seborg, 1997).  To overcome these kinds 
of problems, Ping (1995) and Chen, et al. (2003) 
proposed to use Taylor approximation.  

     The desired trajectory is specified by  a smooth 
vector function x ref(t) ∈ ℜ

n for t∈ [t0, tf]. The problem 
consists in elaborating a control law u(x, t) that 
improves tracking accuracy along the interval [t, t+h], 
where h > 0 is a prediction horizon, such that x( t+h) 
tracks x ref(t+h). That is, the  tracking error is defined by:  

)ht(x)ht(x)ht(e ref +−+=+

)t(u)f(LL
!r

h
)f(L

!r
h

)f(L
2

h
)f(Lh)t(x)ht(x

i
2r

fg
i

r

i
1r

f
i

r

if

2

i
0
fii

i
i

i
i

−− ++

+++=+ L

with fi(x) being the i
th component of f(x) and 

)f(L i
k
f denotes the k

th order Lie derivative of fi with 
respect to f(x). 

[ ]Tn21 )h,x(z)h,x(z)h,x(z)h,x(Z L= ;  

⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛
=

!r
h

!r
h

diag)h(Ë
n

r

1

1r n
L ;  W(x) = [w 1  w2 … wn]

T 

with i
1ri

f
i

ri

if

2

ii fL!r
h

.......fL
2

h
fh)h,x(z −+++=   and  

[ ]i2rifgmi2rif1gi fLLfLLw −−= L  for i=1,…,n. 
2

R

2

Q1
)t(u

2
1

)ht(e
2
1

J ++=

ττ+= ∫ d)t(e2
1

J
h

0

2

Q2

2

R

h

0

2

2Q

2

Q1
)t(u

2
1

d)t(e
2
1

)ht(e
2
1

J +ττ+++= ∫

)t(u)x(W)h(Ë)h,x(Z)t(x)ht(x ++=+

)r(

iref
i

ri

iref

2

irefi

i
x

!r
h

.........x
2

h
xhd +++= &&&



71

The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

The tracking error at the next instant (t+h) is then pre-
dicted as a function of u(t) by:

(10)

(11)

where 

2.2 Tracking Performances
We assume that the matrix W(x) is of full rank. This

assumption is needed for the stability analysis, but not
necessary for the control law to be applicable, since one
can always choose R>0, then the inverse matrix in Eq.
(11) will still exist. Let R=0, then Eq. (11) becomes:

(12)

Let Q1=q1In and Q2=q2In, then the dynamic of the
closed loop equation of the ith component of tracking error
vector e(t) is given in compact form by:

(13)

where

For most mechanical systems with actuator dynamics
neglected, the relative degree is ri=1 or ri=2.  In the case
where ri=1, the eigenvalue of the characteristic equation
of the error dynamic is:

We conclude that the proposed controller achieves faster
tracking error dynamic with regards to Ping, (1995) or
Chen et al. (2003) method. 

3.  Simulation Example

In this section, the reference trajectory-tracking prob-
lem of mobile robot is simulated to show the validity and
achieved performance of the proposed method. A kine-
matics model of a wheeled mobile robot with two degrees
of freedom is given by Kim, et al. (2003):

(14)

where the forward velocity  v and the angular velocity   ω
are considered as the inputs, (x,y) is the center of the rear
axis of vehicle, and  θ is the angle between heading direc-
tion and x-axis. d is the distance from the origin coordi-
nate of the mobile robot to the axis of the driving wheel.

The nonholonomic constraint is written as the following
equation:

(15)

The nonlinear model of the mobile robot can be rewritten
as:

Z = G(θ) U                                                             (16)

where

Note that the above model matches the multi-variable
affine nonlinear system given by Eq. (2) with  f(x) = 0.

Consider the problem of tracking a reference trajecto-
ry given by the equations:

)t(u)x(W)h(Ë)h,t(d

)h,x(Z)t(e)ht(x)ht(x)ht(e ref

+−

+=+−+=+

In order to find the current control u( t) that improves  
tracking er ror along a fixed interval and to avoid the 
computational burden, the expression of the above 
predicted tracking error is used in the objective function 
Eq. (5). Therefore, the unique control signal u op that 

minimizes J, obtained by setting  0
u
J
=

∂
∂

, is (see 

appendix):  

( ))h,x,x(V)t(e)h(K)h(Ã)x(Wu ref11op +−= −−

 Ping’s  method (Ping,1995) : 
h
1

s 0 −= . 

 Chen et al. (2003): 
h2

3
s 0 −=  

 Proposed method:  
h

3
hq

q

2
hq

q
s

2
1

2
1

0

⎟
⎠
⎞

⎜
⎝
⎛

+

⎟
⎠
⎞

⎜
⎝
⎛

+
−= . 

⎪⎩

⎪
⎨
⎧

ω=θ
θ=
θ=

refref

refrefref
refrefref

)sin(vy
)cos(vx

&
&
&

( )

( )dT)T,t(d)T,x(ZQ)T(E

)h,t(d)h,x(ZQ)h(E)h,x,x(V

h

0
2

T

1
T

ref

−+

−=

∫ &&

&&

)t(x)t(x)t(e ref−= ; ∫+=
h

0
21

T dTQ)T(ËQ)h(Ë)h(K ;  

∫+=
h

0
2

TT
1

T dT)T(ËQ)T(Ë)h(ËQ)h(Ë)h(Ã ;  

( )
( ))h,x,x(V)t(e)h(K)x(W

R)x(W)h(A
~

)x(Wu

ref
T

1T
op

+

+−=
−

∑
=

=⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
++

+
ir

0j
i

)j(j

i

2
1 0e!j

h
1rj

hq
q

iref
)j(

i
1j

fi
)j(

xfLe −= −    

for 0 < j < r i and  irefiii
)0(

xxee −==  for j=0.  

     The error dynamics Eq. (13) are linear and time -
invariant. Thus, the proposed controller that minimizes 
the predicted tracking error naturally leads to a special 
case of input/state linearization. The advantage of this 
controller with regards to the linearization method is a 
clear physical meaning of maximum and minimum 
when saturation occurs. Note that, by using Routh -
Criterion, we can show that the tracking error dynamics 
Eq. (13) are stable only for systems with r i ≤ 4.  

θ=θ−θ d)(sinx)(cosy &&

[ ]TyxZ θ= ; 
10

)cos(d)sin(
)sin(d)cos(

)(G θθ
θ−θ

=θ  

and [ ]TvU ω= . 



72

The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

(17)

The optimal control that minimizes the objective function,
Eq. (5), subject to constraints, Eq.  (1) is:

(18)

In the simulation, the control parameters are chosen as: 

h = 0.002,  Q1 = Q2 =500  I3,    R=10-3 I2.

The reference model and initial conditions are:

(19)

The limitations on control signals are:

(20)

Figures 1 and 2 show the tracking trajectory in the xy
plane. Notice how gracefully the mobile robot tracks the
reference trajectory in less than one revolution. Figure 3
depicts the position tracking error given by: 

(21)

We can see that the mobile robot tracks the reference
trajectory successfully. 

It was stated in the introduction that the advantage of
the proposed approach with regard to input-output lin-
earization method is a clear physical meaning of maxi-
mum and minimum control values when saturation
occurs. Figure 4 depicts the control histories v(t) and w(t)
with saturation. Note the initial saturation for both con-
trols due to the large tracking error at the beginning. 

The control parameters Q1, Q2 and R can be chosen as
in classical optimal control theory.  The choice of the con-
trol parameter h, on the other hand, affects the rate of con-
vergence and the amplitude of the control signal.  Indeed,
to bring the control signals inside the saturation limits, one
has to increase the value of h (see Fig. 5) and this will
increase time response of the system slightly. 

4. Conclusions 

In this paper, optimal nonlinear predictive controller
with end point constraints using Taylor approximation is
presented and is applied to multi-variable affine nonlinear
system. Minimizing a quadratic cost function of the pre-
dicted tracking error and the control input derived control

laws. One of the main advantages of this control scheme
is that it does not require an online optimization and
asymptotic tracking of the smooth reference signal is
guaranteed. Moreover, the dynamic of the closed loop sys-
tem is better than the dynamic obtained by both Ping's and
Chen's methods.

The proposed controller is applied to planning motion
problem of mobile robot under nonholonomic constraints.
Finally, we expect that the results presented here can be
explored and extended to discrete implementation of this
continuous-time predictive controller either through com-
puters or special purpose chips that can run at higher
speeds. 

 

⎪⎩

⎪
⎨
⎧

ω=θ
θ=
θ=

refref

refrefref
refrefref

)sin(vy
)cos(vx

&
&
&

⎥
⎦

⎤
⎢
⎣

⎡
⎟
⎠
⎞

⎜
⎝
⎛ +−⎟

⎠
⎞

⎜
⎝
⎛ +θ

⎥
⎦

⎤
⎢
⎣

⎡
+θ⎟

⎠
⎞

⎜
⎝
⎛

+θ−=
−

−

)t(ZQ
3
h

Qh)t(eQ
2
h

Q)(G

Rh)(GQ
3
h

Q)(G
h
1

U

ref2121
T

1
2

21
T

&

where  )t(Z)t(Z)t(e ref−=  
Figure 1. Tracking performance for t ∈ [0, 0.3s] 

Position of the mobile robot and the reference trajectory

y(
t)

x(t)

Figure 2. Tracking performance for t ∈ [0, 2s] 

Position of the mobile robot and the reference trajactory for t = [0.03]

y(
t)

x(t)

ωref = 4rad/s, d=1m, vref =30 m/s, x(0)=0,    
          y(0)= 4 m    and    

2
)0(

π
−=θ  rad. 

smv /500≤    and    sradw /200≤  



73

The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

Boucher, P. and Dumur, D., 1966,   La commande 
prédictive , Collection méthode et pratique de 
l'ingénieur, Edition Technip. Paris. 

Chen, W.H., Balance, D.J. and Gawthrop,  P.J., 2003, 
“Optimal C ontrol of Nonlinear Systems: A 
Predictive C ontrol Approach,” Automatica, Vol. 
39, pp. 633 -641. 

Demircioglu , H. and Gawthrop,  P.J., 1991,  
“Continuous -time Generalized Predictive Control 
(GPC),” Automatica , Vol. 27 (1),  pp. 55-74. 

Henson, M.A. and  Seborg, D.E., 1997,  “Nonlinear 
Process Control,”  Prentice Hall . 

Kim, M.S., Shin, J.H., Hong, S.G. and Lee, J.J., 2003, 
“Designing a Robust Adaptive Dynamic Controller 
for Nonholonomic Mobile Robots under Modeling 
Uncertainty and Disturbances, ” Mechatronics , Vol. 
13, pp. 507 -519. 

Michalska , H. and Mayne, D.Q., 1993,  “Robust 
Receding Horizon Control of Constrained 
Nonlinear S ystems,”  IEEE Transactions on 
Automatic Control , Vol. 38 (11),  pp. 1623 -1633. 

Ping, L., 1995,  “Optimal Predictive Control of 
Continuous Nonlinear Systems,” Int. J. of Control , 
Vol. 62(2), pp. 633 -649. 

Singh, S.N., Steinberg,  M. and  DiGirolamo , R.D., 
1995, “Nonlinear Predictive Control of Feedback 
Linearizable Systems and Flight Control System 
Design,”  J.  of Guidance, Control and Dynamics , 
Vol. 18(5),  pp. 1023 -1028. 

Souroukh, M. and Kravaris,  C., 1996, “ A Continuous-
time Formulation of Nonlinear Model Predictive 
Control,” Int.  J.  of Control , Vol. 63 (1),  pp. 121 -
146. 

References

Figure 3.  Tracking error

Figure 4.  Control signals

Figure 5.  Control signals with h = 0.01 without 
saturation



74

The Journal of Engineering Research  Vol. 3, No. 1 (2006) 69-74

Appendix

By using the predicted tracking error given in Eq. (8), the
cost function can be rewritten:

)t(uR)t(u
2
1

d)t(eQ)t(e
2
1

)ht(eQ)ht(e
2
1

J

T

h

0
2

T
1

T

+

ττ+τ++++= ∫

[ ]T)t(u)x(W)h(Ë)h,t(d)h,x(Ze
2
1

J +−+=

[ ])t(u)x(W)h(Ë)h,t(d)h,x(ZeQ1 +−+

[ ]∫ τ+τ−τ++
h

0

T)t(u)x(W)(Ë),t(d),x(Ze
2
1

[ ] ττ+τ−τ+ d)t(u)x(W)(Ë),t(d),x(ZeQ 2
(t)uR(t)u

2
1 T+

( )

( )

.uR)x(W

d)(EQ)(E)h(EQ)h(E)x(u
2
1

d),t(d),x(ZQ)(E

)h,t(d)h,x(ZQ

)h(EedQ)(EQ)h(E

)x(Wu)h,x,e(MJ

h

0
2

T
1

TTT

h

0
1

T

1

Th

0
2

T
1

T

TT

⎥⎦
⎤+

⎢
⎣

⎡
⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
τττ++

⎥
⎦

⎤
ττ−ττ+

−

⎢
⎣

⎡
+⎟⎟

⎠

⎞
⎜⎜
⎝

⎛
ττ+

+=

∫

∫

∫

&&&&&&&&

&&

&&&&&&
( )

( ))h,x,x(V)t(e)h(K)x(W

R)x(W)h(A
~

)x(Wu

ref
T

1T
op

+

+−=
−

( )

( )dT)T,t(d)T,x(ZQ)T(E

)h,t(d)h,x(ZQ

)h(E)h,x,x(V

h

0
2

T

1

T
ref

−+

−

=

∫ &&

&&

where M(e,x,g) represents terms that are independent of
u(t).

In compact form:

( )

( )uR)x(W)h(Ã)x(Wu
2
1

h,x,x(Ve)h(K)x(Wu)h,x,e(MJ

TT

ref
TT

++

++=

where

)t(x)t(x)t(e ref−= ; ∫+=
h

0
21

T dTQ)T(ËQ)h(Ë)h(K ;   

 ∫+=
h

0
2

T
1

T dT)T(ËQ)T(Ë)h(ËQ)h(Ë)h(Ã ;